# nLab Yoneda lemma for (infinity,1)-categories

Yoneda lemma

## In higher category theory

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Idea

The statement of the Yoneda lemma has a straightforward generalization from categories to (∞,1)-categories.

## Yoneda embedding

###### Definition

For $C$ an (∞,1)-category and $PSh(C)$ its (∞,1)-category of (∞,1)-presheaves, the $(\infty,1)$-Yoneda embedding is the (∞,1)-functor

$y : C \to PSh(C)$

given by $y(X) : U \mapsto C(U,X)$.

## Properties

### Yoneda lemma

###### Theorem

$(\infty,1)$-Yoneda embedding

Let $C$ be an (∞,1)-category and $PSh(C) := Func(C^\op, \infty Grpd)$ be the corresponding (∞,1)-category of (∞,1)-presheaves. Then the canonical (∞,1)-functor

$Y : C \to PSh(C)$
###### Proof

In terms of quasi-categories, this is proposition 5.1.3.1 in

###### Theorem

$(\infty,1)$-Yoneda theorem

For $C$ a small $(\infty,1)$-category and $F : C^{op} \to \infty Grpd$ an $(\infty,1)$-functor, the composite

$C^{op} \to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to} \infty Grpd$

is equivalent to $F$.

###### Proof

This appears as HTT Lemma 5.5.2.1.

The statement is a direct consequence of the sSet-enriched Yoneda lemma by using the fact that the (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(C)$ is modeled by the enriched functor category $[C^{op}, sSet]_{proj}$ with $C$ regarded as a simplicially enriched category and using the global model structure on simplicial presheaves.

### Preservation of limits

###### Theorem

The $(\infty,1)$-Yoneda embedding $y : C \to PSh(C)$ preserves all (∞,1)-limits that exist in $C$.

This appears as HTT, prop. 5.1.3.2.

### Local Yoneda embedding

###### Proposition

For $C$ an (∞,1)-site and $\mathcal{X}$ an (∞,1)-topos, (∞,1)-geometric morphisms $(f^* \dashv f_*) Sh(C) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{X}$ from the (∞,1)-sheaf (∞,1)-topos $Sh(C)$ to $\mathcal{X}$ correspond to the local (∞,1)-functors $f^* : C \to \mathcal{X}$, those that

• are left exact (∞,1)-functors;

• send covering families $\{U_i \to X\}$ in $\mathcal{G}$ to effective epimorphism

$\coprod_i f^*(U_i) \to f^*(X) \,.$

More preseicely, the (∞,1)-functor

$Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G})) \stackrel{L}{\to} Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G})) \stackrel{y}{\to} Func(\mathcal{G}, \mathcal{X})$

given by precomposition of inverse image functors by ∞-stackification and by the (∞,1)-Yoneda embedding is a full and faithful (∞,1)-functor and its essential image is spanned by these local morphisms.

This appears as (HTT, prop. 6.2.3.20).

## References

Published statements appear in

as indicated above.